To make this paper self-contained and to establish notation and terminology, we begin by recalling
the basics of classical joins, strong connections [1]}
and principal comodule algebras [2]}.
In the first section, we define the join and fusion
\(C^*\) -algebra of an arbitrary pair of unital \(C^*\) -algebras. For commutative \(C^*\) -algebras our join
construction recovers
Milnor's definition in the case of compact Hausdorff topological spaces. In the second section, we deal with a
join and fusion of a comodule algebra \(P\)  with the coacting
Hopf algebra \(H\) . To define an \(H\) -comodule algebra structure that corresponds
to the diagonal action in the classical setting,
we provide a “gauged” algebraic version of the join and fusion \(C^*\) -algebras.
(The gauged \(C^*\) -algebraic versions of the join and fusion are given respectively in [3]} and [4]}.)
The main result of this paper is Theorem REF  concluding the principality of the fusion comodule algebra
of \(P\)  and \(H\)  from the principality
of \(P\) .
